Introduction

This analysis examines simulated income data for Hungary, focusing on the relationships between income and various demographic factors including age, location, occupation, and gender. The dataset is simulated to reflect realistic patterns while maintaining a manageable size for analysis.

Data Simulation

The data was simulated by the data_simulation.R script. The data is available in the hungarian_income_data.csv file.

Important things to note about the generated data:

  • Only the 8 most populated cities of Hungary are taken into count weighted by their population. List of cities: Budapest, Debrecen, Szeged, Miskolc, Pécs, GyÅ‘r, Szombathely, Eger.

  • Only the 10 most common occupations are taken into count weighted by their frequency in the workforce. List of occupations: Software Developer, Teacher, Doctor, Sales Representative, Engineer, Accountant, Nurse, Manager, Chef, Driver.

  • The age distribution is generated by a beta distribution with parameters \(\alpha = 2\) and \(\beta = 3\) and multiplied by 95 to put the end result in the desired range. The beta distribution with the aforementioned parameters skews the age distribution towards younger ages, which is more realistic.

  • There are three groups of people categorized by their age:

    • Underage: each person has a random age at which they start working between 14 and 24.

    • Working age: 19-67

    • Pension age: each person has a random retirement age between 60 and 75.

  • Under 18 people have no income.

  • Working age people have a regular income based on their age, occupation, city, and gender.

  • Pension age people have a pension based on their occupation and city.

  • All working age people are considered to be employed.

  • The income of a working age man is 20.000 HUF higher than the income of a working age woman in the same occupation, city, and age group.

Demographics Analysis

library(forcats)

data <- data %>%
  mutate(age_group = cut(age, 
                        breaks = seq(0, 100, by = 2), 
                        right = FALSE, 
                        include.lowest = TRUE, 
                        labels = seq(0, 98, by = 2)))

dem_pyramid <- data %>%
  group_by(age_group, gender) %>%
  summarise(count = n(), .groups = 'drop') %>%
  mutate(count = ifelse(gender == "Male", -count, count))

ggplot(dem_pyramid, aes(x = age_group, y = count, fill = gender)) +
  geom_bar(stat = "identity", width = 0.8, color = "black") +
  scale_y_continuous(labels = abs, expand = expansion(mult = c(0.05, 0.05))) +
  scale_fill_manual(values = c("Male" = "#00BFFF", "Female" = "#FF3B3B")) +
  coord_flip() +
  labs(title = "Population Pyramid of Simulated Hungarian Data",
       x = "Age Group",
       y = "Count",
       fill = "Gender") +
  custom_theme +
  theme(legend.position = "top",
        axis.text.y = element_text(size = 10, face = "bold"),
        plot.margin = margin(t = 20, r = 20, b = 20, l = 20))

Data Preprocessing

We employ two steps to clean the data: inter quartile range (IQR) outlier detection, and clustering.

We use IQR outlier detection to reduce the noise of the data.

We employ K-Means clustering to find the three major demographics groups: unemployed/young, working age, retired.

After cleaning the data and finding the three demographic groups, we solely focus on the middle cluster, the working age people, as this study aims to analyze the income of the Hungarian population and it would be nonsensical to analyze unemployed people or retired people as if their pension was a salary.

detect_outliers <- function(x) {
  q1 <- quantile(x, 0.25)
  q3 <- quantile(x, 0.75)
  iqr <- q3 - q1
  lower_bound <- q1 - 1.5 * iqr
  upper_bound <- q3 + 1.5 * iqr
  return(x < lower_bound | x > upper_bound)
}

outliers <- detect_outliers(data$income)
data_clean <- data[!outliers, ]

set.seed(42)
income_age_matrix <- data_clean %>%
  select(income, age) %>%
  scale()

kmeans_result <- kmeans(income_age_matrix, centers = 3, nstart = 25)

data_clean$cluster <- kmeans_result$cluster

cluster_summary <- data_clean %>%
  group_by(cluster) %>%
  summarise(
    mean_income = mean(income),
    .groups = 'drop'
  ) %>%
  arrange(mean_income)

cluster_labels <- c("Working Age", "Pension Age", "Unemployed/Young")
data_clean$income_group <- factor(data_clean$cluster, 
                                 labels = cluster_labels[order(cluster_summary$mean_income)])

ggplot(data_clean, aes(x = age, y = income, color = income_group)) +
  geom_point(alpha = 0.5) +
  scale_color_viridis_d() +
  labs(title = "Age vs Income by Cluster",
       x = "Age",
       y = "Income (HUF)",
       color = "Income Group") +
  custom_theme

data <- data_clean %>%
  filter(income_group == "Working Age")

Descriptive Statistics

summary_stats <- summary(data)
kable(summary_stats, caption = "Summary Statistics of the Dataset (Outliers Removed)") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)
Summary Statistics of the Dataset (Outliers Removed)
age city occupation gender income starting_age retirement_age age_group cluster income_group
Min. :14.00 Length:7375 Length:7375 Length:7375 Min. :379808 Min. :14.00 Min. :60.00 30 : 397 Min. :1 Working Age :7375
1st Qu.:29.00 Class :character Class :character Class :character 1st Qu.:529234 1st Qu.:18.00 1st Qu.:65.00 32 : 377 1st Qu.:1 Pension Age : 0
Median :39.00 Mode :character Mode :character Mode :character Median :579982 Median :19.00 Median :67.00 34 : 377 Median :1 Unemployed/Young: 0
Mean :40.14 NA NA NA Mean :582750 Mean :18.89 Mean :67.07 28 : 369 Mean :1 NA
3rd Qu.:50.00 NA NA NA 3rd Qu.:632598 3rd Qu.:20.00 3rd Qu.:69.00 42 : 365 3rd Qu.:1 NA
Max. :73.00 NA NA NA Max. :864318 Max. :24.00 Max. :75.00 40 : 360 Max. :1 NA
NA NA NA NA NA NA NA (Other):5130 NA NA

By the following income distribution plot, we can clearly see that on average a man has a higher income than a woman. This does not yet mean that given equal positions a man earns more money. However, it is indicative that we should further analyze this aspect of the data.

ggplot(data %>% filter(age >= 18), aes(x = income, fill = gender)) +
  geom_density(alpha = 0.6) +
  scale_fill_viridis_d() +
  labs(title = "Income Distribution by Gender",
       subtitle = "(working age only)",
       x = "Income (HUF)",
       y = "Density") +
  custom_theme

The following plot shows how the income is distributed against the age. An important thing to note is that as a person ages, their income increase. However it plateaus after a point, moreover, it even decreases in certain cases.

ggplot(data, aes(x = age, y = income, color = gender)) +
  geom_point(alpha = 0.1, width = 0.2) +
  scale_color_viridis_d() +
  labs(title = "Income Distribution by Age",
       subtitle = "(working age only)",
       x = "Age",
       y = "Income (HUF)") +
  custom_theme

To get a better grasp of how the income distribution is made up, we can split the data by occupation, giving us a new perspective into how certain occupation are more handsomely rewarded. We can see that Software Developers and Doctors have the highest income, compared to Sales Representatives who earn a lower income.

ggplot(data %>% filter(age >= 18), aes(x = reorder(occupation, income, FUN = median), y = income, color = occupation)) +
  geom_boxplot(alpha = 0.7) +
  geom_jitter(alpha = 0.1, width = 0.2) +
  scale_color_viridis_d() +
  coord_flip() +
  labs(title = "Income Distribution by Occupation",
       subtitle = "(working age only)",
       x = "Occupation",
       y = "Income (HUF)") +
  custom_theme

The following plot, like the one before, split the data. However, now we are analyzing how the city in which the person works at contributes to their salary. It is hard not to notice that the average person working in the capital, Budapest, enjoys a higher income compared to other cities.

ggplot(data %>% filter(age >= 18), aes(x = reorder(city, income, FUN = median), y = income, fill = city)) +
  geom_violin(alpha = 0.7) +
  geom_boxplot(width = 0.2, alpha = 0.5) +
  scale_fill_viridis_d() +
  coord_flip() +
  labs(title = "Income Distribution by City",
       subtitle = "(working age only)",
       x = "City",
       y = "Income (HUF)") +
  custom_theme

ggplot(data %>% filter(age >= 18), aes(x = age, y = income, color = gender)) +
  geom_point(alpha = 0.3) +
  geom_smooth(method = "loess", se = TRUE) +
  scale_color_viridis_d() +
  labs(title = "Relationship between Age and Income",
       subtitle = "(working age only)",
       x = "Age",
       y = "Income (HUF)") +
  custom_theme

income_by_category <- data %>%
  filter(age >= 18) %>%
  group_by(occupation, city, gender) %>%
  summarise(
    mean_income = mean(income),
    count = n(),
    .groups = 'drop'
  ) %>%
  arrange(desc(mean_income))

# heatmap
ggplot(income_by_category, aes(x = city, y = occupation, fill = mean_income)) +
  geom_tile() +
  scale_fill_viridis(name = "Mean Income (HUF)") +
  facet_wrap(~gender) +
  labs(title = "Mean Income by Occupation, City, and Gender",
       subtitle = "(working age only)",
       x = "City",
       y = "Occupation") +
  custom_theme +
  theme(axis.text.x = element_text(angle = 45, hjust = 1),
        strip.text = element_text(face = "bold"))

top_earners <- income_by_category %>%
  arrange(desc(mean_income)) %>%
  head(10)

kable(top_earners, 
      caption = "Top 10 Highest Earning Combinations",
      digits = 0) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), 
                full_width = FALSE)
Top 10 Highest Earning Combinations
occupation city gender mean_income count
Software Developer Budapest Male 745832 99
Software Developer Budapest Female 722333 100
Doctor Budapest Male 720956 61
Doctor Budapest Female 696091 77
Software Developer Debrecen Male 695662 43
Software Developer Szeged Male 694999 30
Manager Budapest Male 692519 107
Software Developer Szeged Female 684845 43
Engineer Budapest Male 675140 133
Software Developer Debrecen Female 674727 40

Hypothesis Testing

Parametric Tests

1. Gender Income Difference (t-test)

# Test if there's a significant difference in income between genders
t_test_result <- t.test(income ~ gender, data = data)
print(t_test_result)
## 
##  Welch Two Sample t-test
## 
## data:  income by gender
## t = -10.655, df = 7367.6, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group Female and group Male is not equal to 0
## 95 percent confidence interval:
##  -21472.33 -14798.96
## sample estimates:
## mean in group Female   mean in group Male 
##             573558.0             591693.7

2. City Income Differences (ANOVA)

# Test if there are significant differences in income between cities
city_anova <- aov(income ~ city, data = data)
print(summary(city_anova))
##               Df    Sum Sq   Mean Sq F value Pr(>F)    
## city           7 1.356e+13 1.937e+12   539.9 <2e-16 ***
## Residuals   7367 2.643e+13 3.587e+09                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

3. Occupation Income Differences (ANOVA)

# Test if there are significant differences in income between occupations
occupation_anova <- aov(income ~ occupation, data = data)
print(summary(occupation_anova))
##               Df    Sum Sq   Mean Sq F value Pr(>F)    
## occupation     9 1.545e+13 1.717e+12   515.5 <2e-16 ***
## Residuals   7365 2.453e+13 3.331e+09                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Non-parametric Tests

1. Gender Income Distribution (Kolmogorov-Smirnov Test)

# Test if income distributions differ between genders
male_income <- data$income[data$gender == "Male"]
female_income <- data$income[data$gender == "Female"]
ks_test <- ks.test(male_income, female_income)
print(ks_test)
## 
##  Asymptotic two-sample Kolmogorov-Smirnov test
## 
## data:  male_income and female_income
## D = 0.10969, p-value < 2.2e-16
## alternative hypothesis: two-sided

2. Income Distribution by City (Kruskal-Wallis Test)

# Test if income distributions differ between cities
kruskal_test <- kruskal.test(income ~ city, data = data)
print(kruskal_test)
## 
##  Kruskal-Wallis rank sum test
## 
## data:  income by city
## Kruskal-Wallis chi-squared = 2574.8, df = 7, p-value < 2.2e-16

Regression Analysis

Multiple Linear Regression

# Convert categorical variables to factors
data_working_age <- data %>% filter(age >= 18)
data_working_age$city <- as.factor(data_working_age$city)
data_working_age$occupation <- as.factor(data_working_age$occupation)
data_working_age$gender <- as.factor(data_working_age$gender)

# Fit the model
model1 <- lm(income ~ age + I(age^2) + city + occupation + gender, data = data_working_age)

# Create a beautiful model summary table
model_summary <- summary(model1)
kable(tidy(model_summary), caption = "Multiple Linear Regression Results (Working Age Only)") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)
Multiple Linear Regression Results (Working Age Only)
term estimate std.error statistic p.value
(Intercept) 434779.5680 3042.221285 142.91517 0
age 8492.2241 146.411115 58.00259 0
I(age^2) -79.2278 1.733131 -45.71369 0
cityDebrecen -50118.9464 885.873428 -56.57574 0
cityEger -100673.3825 1223.767096 -82.26515 0
cityGyőr -98817.6008 1136.347121 -86.96075 0
cityMiskolc -99847.5434 1064.213618 -93.82284 0
cityPécs -98754.3471 1121.335644 -88.06850 0
citySzeged -49284.8644 997.322532 -49.41718 0
citySzombathely -99991.5016 1348.646293 -74.14212 0
occupationChef -42406.4096 1535.934166 -27.60952 0
occupationDoctor 59742.5726 1554.207886 38.43924 0
occupationDriver -50964.5811 1535.962289 -33.18088 0
occupationEngineer 18051.4651 1251.405236 14.42496 0
occupationManager 37826.6478 1351.616466 27.98623 0
occupationNurse -32257.7329 1187.823213 -27.15702 0
occupationSales Representative -60650.2152 1057.574345 -57.34842 0
occupationSoftware Developer 88380.8010 1335.055549 66.20009 0
occupationTeacher -22511.0613 1128.935083 -19.94008 0
genderMale 18870.6304 582.757427 32.38162 0
# Enhanced model diagnostics
par(mfrow = c(2,2))
plot(model1, col = income_palette[1], pch = 19, cex = 0.7)

Polynomial Regression for Age-Income Relationship

# Fit polynomial regression
model2 <- lm(income ~ poly(age, 3), data = data_working_age)

# Create a static plot
ggplot(data_working_age, aes(x = age, y = income)) +
  geom_point(alpha = 0.1, color = income_palette[1]) +
  geom_smooth(method = "lm", formula = y ~ poly(x, 3), 
              color = income_palette[5], fill = income_palette[5], alpha = 0.2) +
  labs(title = "Polynomial Regression: Age vs Income",
       subtitle = "Cubic polynomial fit with confidence interval (Working Age Only)",
       x = "Age",
       y = "Income (HUF)") +
  custom_theme

# Model comparison
model_comparison <- data.frame(
  Model = c("Multiple Linear", "Polynomial"),
  R_squared = c(summary(model1)$r.squared, summary(model2)$r.squared),
  Adj_R_squared = c(summary(model1)$adj.r.squared, summary(model2)$adj.r.squared)
)

kable(model_comparison, caption = "Model Comparison (Working Age Only)") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)
Model Comparison (Working Age Only)
Model R_squared Adj_R_squared
Multiple Linear 0.8851681 0.8848691
Polynomial 0.1541084 0.1537614

Predictions

What is the predicted income of a 35 years old male software developer working in Budapest?

# Create a sample prediction
new_data <- data.frame(
  age = 35,
  city = "Budapest",
  occupation = "Software Developer",
  gender = "Male"
)

# Predict income
prediction <- predict(model1, newdata = new_data, interval = "prediction")
print(prediction[1])
## [1] 742204.8

Summary and Conclusions

The analysis of the simulated Hungarian income data reveals several interesting patterns:

  1. There is a significant gender pay gap, with males earning more on average than females.
  2. Income varies significantly across different cities, with Budapest showing the highest average income.
  3. Occupation has a strong effect on income, with software developers and doctors earning the most.
  4. Age shows a non-linear relationship with income, peaking in the middle age range.
  5. The regression models explain a significant portion of the income variation.

The analysis demonstrates the complex interplay between various factors affecting income levels in Hungary. The findings suggest that both demographic factors (age, gender) and professional factors (occupation, location) play important roles in determining income levels.